{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:59:49Z","timestamp":1760241589470,"version":"build-2065373602"},"reference-count":66,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2018,6,5]],"date-time":"2018-06-05T00:00:00Z","timestamp":1528156800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Climate has complex dynamics due to the plethora of phenomena underlying its evolution. These characteristics pose challenges to conducting solid quantitative analysis and reaching assertive conclusions. In this paper, the global temperature time series (TTS) is viewed as a manifestation of the climate evolution, and its complexity is calculated by means of four different indices, namely the Lempel\u2013Ziv complexity, sample entropy, signal harmonics power ratio, and fractal dimension. In the first phase, the monthly mean TTS is pre-processed by means of empirical mode decomposition, and the TTS trend is calculated. In the second phase, the complexity of the detrended signals is estimated. The four indices capture distinct features of the TTS dynamics in a 4-dim space. Hierarchical clustering is adopted for dimensional reduction and visualization in the 2-dim space. The results show that TTS complexity exhibits space-time variability, suggesting the presence of distinct climate forcing processes in both dimensions. Numerical examples with real-world data demonstrate the effectiveness of the approach.<\/jats:p>","DOI":"10.3390\/e20060437","type":"journal-article","created":{"date-parts":[[2018,6,5]],"date-time":"2018-06-05T04:16:43Z","timestamp":1528172203000},"page":"437","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Complexity Analysis of Global Temperature Time Series"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7359-4370","authenticated-orcid":false,"given":"Ant\u00f3nio M.","family":"Lopes","sequence":"first","affiliation":[{"name":"UISPA\u2013LAETA\/INEGI, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200\u2013465 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4274-4879","authenticated-orcid":false,"given":"J. A.","family":"Tenreiro Machado","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, R. Dr. Ant\u00f3nio Bernardino de Almeida, 431, 4249\u2013015 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2018,6,5]]},"reference":[{"key":"ref_1","unstructured":"Solomon, S., Qin, D., Manning, M., Chen, Z., Marquis, M., Averyt, K.B., Tignor, M., and Miller, H.L. (2007). Contribution of Working Group I to the Fourth Assessment Report Of the Intergovernamental Panel on Climate Change, Cambridge University Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1038\/nature01333","article-title":"Fingerprints of global warming on wild animals and plants","volume":"421","author":"Root","year":"2003","journal-title":"Nature"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1038\/nclimate3275","article-title":"Drylands face potential threat under 2 \u2218C global warming target","volume":"7","author":"Huang","year":"2017","journal-title":"Nat. Clim. Chang."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1002\/2016EF000410","article-title":"Arctic ice management","volume":"5","author":"Desch","year":"2017","journal-title":"Earth\u2019s Future"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1016\/j.cnsns.2013.08.031","article-title":"Analysis of temperature time-series: Embedding dynamics into the MDS method","volume":"19","author":"Lopes","year":"2014","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/j.ecolmodel.2006.11.020","article-title":"Global warming and human activity: A model for studying the potential instability of the carbon dioxide\/temperature feedback mechanism","volume":"203","author":"Alexiadis","year":"2007","journal-title":"Ecol. Model."},{"key":"ref_7","unstructured":"Working Group, I. (2001). The Scientific Basis. Climate Change, IPCC. Third Assessment Report of the Intergovernamental Panel on Climate Change."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1002\/wcc.81","article-title":"Drought under global warming: A review","volume":"2","author":"Dai","year":"2011","journal-title":"Wiley Interdiscip. Rev. Clim. Chang."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.gloplacha.2010.04.003","article-title":"Climate warming and associated changes in atmospheric circulation in the eastern and central Tibetan Plateau from a homogenized dataset","volume":"72","author":"You","year":"2010","journal-title":"Glob. Planet. Chang."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.gloplacha.2011.09.006","article-title":"Sea level projections to AD2500 with a new generation of climate change scenarios","volume":"80","author":"Jevrejeva","year":"2012","journal-title":"Glob. Planet. Chang."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/j.gloplacha.2009.06.001","article-title":"Climatic changes and associated impacts in the Mediterranean resulting from a 2 \u2218C global warming","volume":"68","author":"Giannakopoulos","year":"2009","journal-title":"Glob. Planet. Chang."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1463","DOI":"10.1007\/s00382-010-0852-9","article-title":"Recent changes in the summer precipitation pattern in East China and the background circulation","volume":"36","author":"Zhu","year":"2011","journal-title":"Clim. Dyn."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"RG4004","DOI":"10.1029\/2010RG000345","article-title":"Global surface temperature change","volume":"48","author":"Hansen","year":"2010","journal-title":"Rev. Geophys."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Brohan, P., Kennedy, J.J., Harris, I., Tett, S.F., and Jones, P.D. (2006). Uncertainty estimates in regional and global observed temperature changes: A new data set from 1850. J. Geophys. Res. Atmos., 111.","DOI":"10.1029\/2005JD006548"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1700","DOI":"10.1175\/2008JCLI2263.1","article-title":"Homogenization of temperature series via pairwise comparisons","volume":"22","author":"Menne","year":"2009","journal-title":"J. Clim."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1007\/s00382-011-1128-8","article-title":"On the time-varying trend in global-mean surface temperature","volume":"37","author":"Wu","year":"2011","journal-title":"Clim. Dyn."},{"key":"ref_17","first-page":"1","article-title":"Berkeley Earth temperature averaging process","volume":"1","author":"Rohde","year":"2013","journal-title":"Geoinf. Geostat. Overv."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Lawrimore, J.H., Menne, M.J., Gleason, B.E., Williams, C.N., Wuertz, D.B., Vose, R.S., and Rennie, J. (2011). An overview of the Global Historical Climatology Network monthly mean temperature data set, version 3. J. Geophys. Res. Atmos., 116.","DOI":"10.1029\/2011JD016187"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Jones, P., Lister, D., Osborn, T., Harpham, C., Salmon, M., and Morice, C. (2012). Hemispheric and large-scale land-surface air temperature variations: An extensive revision and an update to 2010. J. Geophys. Res. Atmos., 117.","DOI":"10.1029\/2011JD017139"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/j.gloplacha.2007.10.001","article-title":"Time series analysis and identification of trends in a Mediterranean urban area","volume":"63","author":"Capilla","year":"2008","journal-title":"Glob. Planet. Chang."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/s007040200003","article-title":"Statistical time series decomposition into significant components and application to European temperature","volume":"71","author":"Grieser","year":"2002","journal-title":"Theor. Appl. Climatol."},{"key":"ref_22","first-page":"241","article-title":"Statistical analysis and time-series models for minimum\/maximum temperatures in the Antarctic Peninsula","volume":"463","author":"Hughes","year":"2007","journal-title":"Proc. R. Soc. Lond. A Math. Phys. Eng. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1964","DOI":"10.1016\/j.ecolmodel.2010.05.001","article-title":"Analysis of the global warming dynamics from temperature time series","volume":"221","author":"Viola","year":"2010","journal-title":"Ecol. Model."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.gloplacha.2004.06.003","article-title":"Analysis of mean, maximum, and minimum temperature in Athens from 1897 to 2001 with emphasis on the last decade: Trends, warm events, and cold events","volume":"44","author":"Founda","year":"2004","journal-title":"Glob. Planet. Chang."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Ge, Q.S., Zheng, J.Y., Hao, Z.X., Shao, X.M., Wang, W.C., and Luterbacher, J. (2010). Temperature variation through 2000 years in China: An uncertainty analysis of reconstruction and regional difference. Geophys. Res. Lett., 37.","DOI":"10.1029\/2009GL041281"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Deser, C., Phillips, A.S., and Alexander, M.A. (2010). Twentieth century tropical sea surface temperature trends revisited. Geophys. Res. Lett., 37.","DOI":"10.1029\/2010GL043321"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1002\/(SICI)1097-0088(199607)16:7<805::AID-JOC48>3.0.CO;2-Z","article-title":"Temperature variations in Spain since 1901: A preliminary analysis","volume":"16","author":"Pou","year":"1996","journal-title":"Int. J. Climatol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"300","DOI":"10.3402\/tellusa.v52i3.12267","article-title":"Statistical methods for interpreting Monte Carlo ensemble forecasts","volume":"52","author":"Stephenson","year":"2000","journal-title":"Tellus A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1775","DOI":"10.3390\/e17041775","article-title":"Multidimensional scaling visualization using parametric similarity indices","volume":"17","author":"Machado","year":"2015","journal-title":"Entropy"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1109\/TIT.1976.1055501","article-title":"On the complexity of finite sequences","volume":"22","author":"Lempel","year":"1976","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"4143","DOI":"10.1038\/s41598-017-04584-x","article-title":"Contrasting the complexity of the climate of the past 122,000 years and recent 2000 years","volume":"7","author":"Shao","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1007\/s40846-016-0165-5","article-title":"Using Lempel\u2013Ziv complexity to assess ECG signal quality","volume":"36","author":"Zhang","year":"2016","journal-title":"J. Med. Biol. Eng."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2742","DOI":"10.1152\/jn.00575.2014","article-title":"Lempel-Ziv complexity of cortical activity during sleep and waking in rats","volume":"113","author":"Simons","year":"2015","journal-title":"J. Neurophysiol."},{"key":"ref_34","first-page":"103","article-title":"Complexity and brain function","volume":"7","author":"Wu","year":"1991","journal-title":"Acta Biophy. Sin."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1109\/51.677174","article-title":"Estimating regularity in epileptic seizure time-series data","volume":"17","author":"Radhakrishnan","year":"1998","journal-title":"IEEE Eng. Med. Biol. Mag."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1007\/s10910-010-9673-7","article-title":"A generalization of Lempel-Ziv complexity and its application to the comparison of protein sequences","volume":"48","author":"Li","year":"2010","journal-title":"J. Math. Chem."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1371","DOI":"10.1109\/TBME.2002.804582","article-title":"Quantifying physiological data with Lempel-Ziv complexity-certain issues","volume":"49","author":"Nagarajan","year":"2002","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2606","DOI":"10.1109\/TBME.2006.883825","article-title":"Analysis of biomedical signals by the Lempel-Ziv complexity: The effect of finite data size","volume":"53","author":"Hu","year":"2006","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"548","DOI":"10.1109\/10.759055","article-title":"Detecting ventricular tachycardia and fibrillation by complexity measure","volume":"46","author":"Zhang","year":"1999","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2282","DOI":"10.1109\/TBME.2006.883696","article-title":"Interpretation of the Lempel-Ziv complexity measure in the context of biomedical signal analysis","volume":"53","author":"Aboy","year":"2006","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"023306","DOI":"10.1103\/PhysRevE.91.023306","article-title":"Analysis of the phase transition in the two-dimensional Ising ferromagnet using a Lempel-Ziv string-parsing scheme and black-box data-compression utilities","volume":"91","author":"Melchert","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"2297","DOI":"10.1073\/pnas.88.6.2297","article-title":"Approximate entropy as a measure of system complexity","volume":"88","author":"Pincus","year":"1991","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1063\/1.166092","article-title":"Approximate entropy (ApEn) as a complexity measure","volume":"5","author":"Pincus","year":"1995","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/j.chaos.2015.09.002","article-title":"Complexity testing techniques for time series data: A comprehensive literature review","volume":"81","author":"Tang","year":"2015","journal-title":"Chaos Solitons Fractals"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"H2039","DOI":"10.1152\/ajpheart.2000.278.6.H2039","article-title":"Physiological time-series analysis using approximate entropy and sample entropy","volume":"278","author":"Richman","year":"2000","journal-title":"Am. J. Physiol. Heart Circ. Physiol."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1026","DOI":"10.1016\/j.nonrwa.2009.01.047","article-title":"A novel application of sample entropy to the electrocardiogram of atrial fibrillation","volume":"11","author":"Alcaraz","year":"2010","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"043105","DOI":"10.1063\/1.4758815","article-title":"Evaluation of physiologic complexity in time series using generalized sample entropy and surrogate data analysis","volume":"22","year":"2012","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.physa.2013.09.062","article-title":"Complexity analysis of the turbulent environmental fluid flow time series","volume":"395","year":"2014","journal-title":"Phys. Stat. Mech. Its Appl."},{"key":"ref_49","unstructured":"Stein, E.M., and Shakarchi, R. (2003). Fourier Analysis: An Introduction, Princeton University Press."},{"key":"ref_50","unstructured":"Dym, H., and McKean, H. (1972). Fourier Series and Integrals, Academic Press."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1098\/rspa.1998.0193","article-title":"The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis","volume":"454","author":"Huang","year":"1998","journal-title":"Proc. Math. Phys. Eng. Sci."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1155\/2008\/293056","article-title":"Discrimination between ictal and seizure-free EEG signals using empirical mode decomposition","volume":"2008","author":"Pachori","year":"2008","journal-title":"Res. Lett. Signal Process."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"14889","DOI":"10.1073\/pnas.0701020104","article-title":"On the trend, detrending, and variability of nonlinear and nonstationary time series","volume":"104","author":"Wu","year":"2007","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"781","DOI":"10.1088\/0305-4470\/12\/6\/008","article-title":"Diffractals","volume":"12","author":"Berry","year":"1979","journal-title":"J. Phys. Math. Gen."},{"key":"ref_55","first-page":"171","article-title":"Tambour fractal: Vers une r\u00e9solution de la conjecture de Weyl-Berry pour les valeurs propres du laplacien","volume":"306","author":"Lapidus","year":"1988","journal-title":"C. R. Acad. Sci. S\u00e9r. Math."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Val\u00e9rio, D., Lopes, A.M., and Tenreiro Machado, J.A.T. (2016). Entropy Analysis of a Railway Network\u2019s Complexity. Entropy, 18.","DOI":"10.3390\/e18110388"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"897","DOI":"10.1175\/JTECH-D-11-00103.1","article-title":"An overview of the global historical climatology network-daily database","volume":"29","author":"Menne","year":"2012","journal-title":"J. Atmos. Ocean. Technol."},{"key":"ref_58","unstructured":"Steffen, K., Box, J., and Abdalati, W. (1996). Greenland Climate Network: GC-Net, CRREL. US Army Cold Regions Reattach and Engineering (CRREL), CRREL Special Report."},{"key":"ref_59","unstructured":"Willmott, C.J., and Matsuura, K. (2018, February 20). Terrestrial Air Temperature and Precipitation: Monthly and Annual Time Series (1950\u20131996). Available online: http:\/\/climate.geog.udel.edu\/~climate\/html_pages\/Global2014\/README.GlobalTsT2014.html."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"2317","DOI":"10.1098\/rspa.2003.1123","article-title":"A confidence limit for the empirical mode decomposition and Hilbert spectral analysis","volume":"459","author":"Huang","year":"2003","journal-title":"Proc. R. Soc. Math. Phys. Eng. Sci."},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Polderman, J.W., and Willems, J.C. (1998). Introduction to Mathematical Systems Theory: A Behavioral Approach, Springer.","DOI":"10.1007\/978-1-4757-2953-5"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"2153","DOI":"10.1177\/1077546314565687","article-title":"State space analysis of forest fires","volume":"22","author":"Lopes","year":"2016","journal-title":"J. Vib. Control"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"3892","DOI":"10.3390\/e15093892","article-title":"Analysis and visualization of seismic data using mutual information","volume":"15","author":"Machado","year":"2013","journal-title":"Entropy"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1093\/comjnl\/26.4.354","article-title":"A survey of recent advances in hierarchical clustering algorithms","volume":"26","author":"Murtagh","year":"1983","journal-title":"Comput. J."},{"key":"ref_65","first-page":"300","article-title":"Comprehensive survey on distance\/similarity measures between probability density functions","volume":"4","author":"Cha","year":"2007","journal-title":"Int. J. Math. Models Methods Appl. Sci."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1175\/EI-D-15-0038.1","article-title":"Towards an understanding of the twentieth-century cooling trend in the southeastern United States: Biogeophysical impacts of land-use change","volume":"20","author":"Ellenburg","year":"2016","journal-title":"Earth Interact."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/6\/437\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:07:20Z","timestamp":1760195240000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/6\/437"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6,5]]},"references-count":66,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2018,6]]}},"alternative-id":["e20060437"],"URL":"https:\/\/doi.org\/10.3390\/e20060437","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2018,6,5]]}}}